America “deserves to hear the truth about possible Russian involvement in our elections,” the Republican chairman of the Senate Intelligence Committee said Thursday, as the panel opened its first hearings on what members predict will be a probe of Moscow’s activities likely to last months or more.

“How they came to be involved, how we may have failed to prevent that involvement, what actions were taken in response, if any, and what we plan to do to ensure the integrity of future, free elections at the heart of our democracy” will all be on the table, said Sen. Richard Burr of North Carolina.

His comments were buttressed by similar statements from Sen. Mark Warner of Virginia, the panel’s ranking Democrat, in a calculated move by the Senate to try to inject sobriety into an investigation that has found itself at the center a media firestorm over the extent to which President Trump or any of his associates may have known of Russia’s meddling prior to the 2016 election.

For a second day in a row, Mr. Burr and Mr. Warner sought to tamp down partisan backbiting that has derailed a parallel House side probe over accusations that House Intelligence Committee Chairman Devin Nunes colluded with the White House on the probe of U.S. spy agencies.

Mr. Warner said the coming investigation will grow out of a foundation laid out by a U.S. intelligence community assessment circulated in the closing days of the Obama administration that said Russian hacking and propaganda had aided Mr. Trump’s presidential bid.

While Mr. Warner said he and Mr. Burr “agree it is vitally important that we do this in as credible, bipartisan, and transparent a manner as possible,” the very nature of the intelligence community assessment at hand could make that an insurmountable task.

While Mr. Warner said hacking and subsequent propaganda by Russian operatives were “seemingly choreographed” to “cause maximum damage to one candidate,” he took care to assert that the “unprecedented level of sophistication” behind Moscow’s actions “should be a cause of concern for everyone.”

The sentiment was underscored by Sen. Marco Rubio of Florida, a former Republican presidential candidate who revealed Thursday afternoon that his own political operations were the target of what appeared to be Russian hacking — as recently as this week.

“In July of 2016,” Mr. Rubio said, “former members of my presidential campaign team, who had access to the internal information of my presidential campaign, were targeted by IP addresses with an unknown location within Russia. That effort was unsuccessful.”

“Within the last 24 hours, at 10:45 a.m. [Wednesday], a second attempt was made, again against former members of my presidential campaign team…[And] again targeted from an IP address from an unknown location in Russia,” Mr. Rubio said. “That effort was also unsuccessful.”

Mr. Burr, meanwhile, said the committee’s investigation “will fail” if members let politics get in the way.

The North Carolina Republican has put his weight behind a plan begin the Senate probe by questioning 20 people, including President Trump’s son-in-law and advisor Jared Kushner.

It is not yet clear when Mr. Kushner will testify. But lawmakers have asked him to testify on grounds he met Russia’s ambassador to Washington in December and as well as the head of Russia’s state-owned development bank.

Administration supporters say the meetings were normal engagements for an advisor to a leading presidential campaign, but critics claim they could be the tip of a conspiracy between Mr. Trump’s associates and Moscow.

Thursday’s hearings had a less focused tenor, with testimony from a collection of cybersecurity and Russian history experts, as well as retired Army Gen. Keith Alexander, who served until 2014 as director of the National Security Agency and as the first chief of U.S. Cyber Command.

Roy Godson, a Georgetown University professor and long-time national security consultant, told the panel that Russian meddling is occurring not just in the U.S., but around the world. President Vladimir Putin has sought to foster in an era of clandestine subversion akin to what the Soviet Union engaged in at the height of its power a half-century ago, he said.

Mr. Putin, whose own background is in Soviet KGB intelligence, came to the fore in 1999 with “a coterie of former colleagues, many also trained in the Soviet security and intelligence system,” Mr. Godson said, adding the group is bent on “discrediting the U.S. and democratic society in general.”

“Their focus is almost completely negative, zeroing in on creating chaos and division in what has been called an ‘age of anger’ in many parts of the world,” he said. “This opens up many opportunities for influence.”

Mr. Alexander told lawmakers Russia’s international subversion operations date back to the 1920s and noted how one-time KGB Maj. Gen. Oleg Kalugin described them as “the heart and soul of the Soviet intelligence,” specifically designed to “weaken the West, to drive wedges in the Western community alliances of all sorts, particularly NATO, [and] to dow discord among allies.”